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I'm reading Kuyk's book about the foundation of mathematics and its history. Talking about Plato, he says that

...But let us follow the track of the Greeks and their philosophy of mathematics. The next important development after Pythagoreanism in the philosophy of mathematics is that which Plato (428/7 - 348/7 B.C.) introduced. We can do no better than to quote one paragraph from Aristotle'sMetaphysics as to what this introduction was, how it related to what had gone on previous to Plato, and what an inherent difficulty lay in this position.

For, having in his (= Plato's) youth first become familiar with Cratylos and with the Herac1itean doctrines (that a11 sensible things are ever in a state of flux and there is no knowledge about them), these views he held even in later years. Socrates, however, was busying himself about ethical matters and neglecting the world of nature as a whole but seeking the universal in these ethical matters, and fixed thought for the first time on definitions; Plato accepted his teaching, but held that the problem applied not to sensible things but to entities of another kind - for this reason, that the common definition could not be a definition of any sensible thing, as they were always changing. Things of this other sort, then, he called IDEAS, and sensible things, he said, were a11 named after these, and in virtue of a relation to these; for the many existed by participation in the Ideas that have the same name as they. Only the name 'participation' was new; for the Pythagoreans say that things exist by 'imitation' of numbers, and Plato says they exist by participation, changing the name. But wh at the participation or imitation of the Forms could be they left an open question. (Book A, Chapter 1, 987 a 31 - b 13)

It is clear from this quotation that Plato was concerned with the foundation of true knowledge, which foundation he -and all the pre-Socratic thinkers - sought in the nature of things (ontology).

I know this question is very basic, but I'm very new to philosophy. According to the context, what does Kuyk mean by the term foundation in the last paragraph?

Thank you in advance for any guide.

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    There is a term "foundation" in epistemology that stands for basic knowledge that does not need to be further justified. But, from context, Kuyk uses the word colloquially, not for basic or any knowledge itself but for its basis in reality, what supports it, what makes the true knowledge true, called its truthmaker. Some pre-Socratics sought it in the sensible things, but Heraclitus's flux put that in doubt. Flux cannot support anything firm or permanent. So Plato shifted truthmaking from fleeting sensible things to everlasting Ideas.
    – Conifold
    Commented Aug 25 at 0:36
  • i think anyone who is fluent in english should understand the passage with no issues whatsoever. do you know what the word foundation means?
    – Dennis Kozevnikoff
    Commented Aug 25 at 13:55
  • See Plato on Knowledge in the Theaetetus: the Theaetetus offers a set-piece discussion of the question “What is knowledge?” Theaetetus’ third proposal about how to knowledge is that it is true belief with an account (meta logou alêthê doxan). The proposed account of logos says that to give the logos of O is to cite the sêmeion or diaphora of O. [...] it means the ‘sign’ or diagnostic feature wherein x differs from everything else, or everything else of O’s own kind. 1/2
    – Mauro ALLEGRANZA
    Commented Aug 26 at 7:14
  • So, presumably, knowledge of (say) Theaetetus consists in true belief about Theaetetus plus an account of what differentiates Theaetetus from every other human. On this conception, knowledge will come about when someone is capable not only of using such logical constructions in thought, but of understanding how they arise from perception. 2/2
    – Mauro ALLEGRANZA
    Commented Aug 26 at 7:14

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I would say the foundation of knowledge (form the Greek pov) is sensation, as an axiom, and then the connection or correlation of different sensations to create relationships.

For example, you see your mother and feel her caress at the same time. "Mother exists" as a start of knowledge.

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